Math, to me, has always been about puzzles — and I love doing puzzles. As a kid, I used to spend hours doing puzzle books for fun. Even now, when I approach a complex problem, I still think about it as if it were a game. So I thought I would share three of my favorite puzzles, in increasing order of difficulty. A word of warning: I’m not including answers. When I’m working on a math problem, I have to understand my work well enough to convince myself that it’s right. Math doesn’t have answers in the back of the puzzle book. Neither does life. Enjoy.
1. Football Geometry
This list would not be complete without a football puzzle. Suppose there is a taut 120-yard rope tied to the base of each goalpost. How much slack would need to be added so that a man who stands six feet tall could walk underneath it? How much so that a three-story (30-foot) house could fit underneath it?
2. Saint Petersburg Paradox
Our second puzzle involves gambling. Let me describe a new game at the casino. The rules are as follows:
The house puts one dollar in a pot. The dealer flips a coin. If the coin comes up heads, the house doubles the amount of money in the pot and the dealer flips the coin again. If the coin comes up tails, then the game is over and you win the contents of the pot. The game costs X dollars to play.
If the casino is to break even playing this game, how much should they charge — what should be the value of X? And how much would you be willing to pay to play this game?
3. Einstein’s Puzzle
This puzzle is named after Albert Einstein because he was rumored to have given it to prospective advisees as a way to determine the most dedicated and best students. According to Einstein only 2% of the population could solve the puzzle. Are you one of them?
Five people, each of a different nationality, live in five adjacent houses, each of a different color. Each person owns a different pet, drinks a different beverage and smokes a different type of cigarette. You have 15 pieces of information (see below) about the group. It is up to you to deduce who owns the fish!
1. The Englishman lives in the red house.
2. The Swede keeps dogs.
3. The Dane drinks tea.
4. The green house is just to the left of the white one.
5. The owner of the green house drinks coffee.
6. The Pall Mall smoker keeps birds.
7. The owner of the yellow house smokes Dunhills.
8. The man in the center house drinks milk.
9. The Norwegian lives in the first house.
10. The blend smoker has a neighbor who keeps cats.
11. The man who smokes Blue Masters drinks beer.
12. The man who keeps horses lives next to the Dunhill smoker.
13. The German smokes Prince cigarettes.
14. The Norwegian lives next to the blue house.
15. The blend smoker has a neighbor who drinks water.
(And if you’re really torn up about the answers to these puzzles, you can shoot me a tweet @JohnCUrschel and I might just help you out.)